Let f be a polynomial with coefficients in a field K and let t be a variable. The Galois group of the extension of K(t) formed by adjoining all iterated f pre-images of t is called the arithmetic profinite iterated monodromy group (pIMG) of f. I'll discuss some recent joint work with Ophelia Adams on the structure of the pIMG of when f is a unicritical polynomial with degree coprime to char(K).